F 

1563 
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B38 


BECKER 

MECHANICS  OF  THE 
PANAMA  CANAL  SLIDES 


BANCROFT 


wm^- 

University  of  California  •  Berkeley 


DEPARTMENT  OF  THE  INTERIOR 

FRANKLIN  K.  LANE,  Secretary 


UNITED  STATES  GEOLOGICAL  SURVEY 

GEORGE  OTIS  SMITH,  Director 


PROFESSIONAL  PAPER  98— N 


MECHANICS  OF  THE  PANAMA  CANAL  SLIDES 


BY 


GEORGE  F.  BECKER 


Published  July  25, 1916 


Shorter  contributions  to  general  geology,  1916 

(Pages  253-261) 


WASHINGTON 

GOVERNMENT    PRINTING    OFFICE 

1916 


DEPARTMENT  OF  THE  INTERIOR    . 

FRANKLIN  K.  LANE,  Secretary 


UNITED  STATES  GEOLOGICAL  SURVEY 

GEORGE  OTIS  SMITH,  Director 


Professional  Paper  98 — N 


MECHANICS  OP  THE  PANAMA  CANAL  SLIDES 


BY 

GEORGE  F.  BECKER 


Published  July  25,  1916 


Shorter  contributions  to  general  geology,  1916 

(Pages  253-261) 


WASHINGTON 

GOVKKNMENT    PRINTING    OFFICE 
1916 


CONTENTS. 


Page. 

Prefatory  note,  by  George  Otis  Smith 253 

Observations  on  the  slides 253 

Limiting  depth  of  disturbance 254 

Conditions  in  a  wide  cut 254 

Limiting  values  of  k 256 

Examples  of  slide  curves 256 

Hydrostatic  analogy 257 

Formation  of  ruptures 258 

Bulging  of  canal  bottom 258 

Effect  of  the  form  of  the  banks 258 

Note  on  finite  strains 259 

Summary 261 


ILLUSTRATIONS. 


Page. 
FIGURE  22.  Curve  of  uniform  tangential  strain 255 

23.  Elastic  curves  for  a=75°,  80°,  85°,  89° 257 

24.  Diagram  illustrating  simple  shear  and  shear ,  each  of  ratio  5/4 260 


MECHANICS  OF  THE  PANAMA  CANAL  SLIDES. 


By  GEORGE  F.  BECKER. 


PREFATORY  NOTE. 

By  GEORGE  OTIS  SMITH. 

This  geophysical  study  of  the  Panama  Canal 
slides  is  now  presented  for  the  reasons  set  forth 
in  the  following  letter  of  transmittal: 

The  DIRECTOR. 
SIR: 

I  have  the  honor  to  transmit  herewith  a  paper  on  the 
mechanics  of  the  Panama  Canal  elides.     It  was  prepared 
as  a  contribution  to  the  report  of  the  committee  of  the 
National  Academy  of  Sciences  on  the  Panama  Canal  slides, 
appointed  at  the  instance  of  the  President  of  the  United 
States. 

As  some  delay1  is  anticipated  in  completing  the  full  re- 
port, this  chapter  is  now  submitted  for  publication  with 
the  sanction  of  President  C.  R.  Van  Hise,  chairman  of  the 
committee. 

Very  truly,  yours, 

GEORGE  F.  BECKER. 

Dr.  Becker  visited  the  Canal  Zone  in  1913  as 
a  geologist  of-  the  United  States  Geological 
Survey  and  since  that  time  has  given  the  prob- 
lem the  benefit  of  his  study.  His  appointment 
as  a  member  of  the  committee  of  the  National 
Academy  of  Sciences  has  made  it  appropriate 
for  his  conclusions,  based  upon  his  personal 
observations  and  already  reported  in  part  to 
the  Canal  Commission,  to  be  stated  for  the 
benefit  of  his  associates  and  other  American 
scientists  and  engineers. 

OBSERVATIONS  ON  THE  SLIDES. 

Early  in  1913,  before  water  was  admitted,  I 
spent  some  weeks  hi  examining  the  geology 
of  the  Culebra  Cut,  now  officially  known  as  the 
Gaillard  Cut,  with  special  reference  to  the  origin 
of  the  landslides.1  These  appear  to  me  to  be 
of  two  kinds — mere  superficial  slips  on  joint 
planes  or  other  slippery  surfaces  and  deeper- 
seated  "breaks,"  as  they  are  known  by  the 

I 1  had  the  great  advantage  of  Mr.  Donald  MacDonald's  companion- 
ship throughout  these  field  studies. 

45498°— 16 


engineers.  It  is  only  with  the  latter  that  this 
paper  is  concerned. 

The  breaks  in  their  inception  are  marked  on 
comparatively  level  banks  by  groups  of  cracks 
or  narrow  fissures  nearly  parallel  to  the  cut, 
and  these  almost  immediately  develop  into 
series  of  step  faults  with  small  throws,  many  of 
them  only  a  fraction  of  an  inch  hi  height,  the 
hade  where  not  vertical  being  invariably  to- 
ward the  canal  so  far  as  I  could  observe.2  Many 
of  the  steps  of  these  faults  are  only  a  yard  or 
two  in  width.  There  seems  little  order  in  the 
time  of  formation  of  the  cracks;  in  some  breaks 
groups  of  small  faults  first  appear  rather  close 
to  the  cut,  those  at  a  greater  distance  from  it 
developing  later.  In  others  the  earliest  cracks 
are  hundreds  of  feet  from  the  canal  and  the 
intermediate  ground  splits  up  afterward.  In 
all  the  breaks  which  I  could  examine  the  first 
small  movements  involved  no  perceptible  gap- 
ing, or  none  of  the  same  order  of  magnitude  as 
the  throws  of  the  faults.  At  or  about  the  same 
time  as  the  cracks  on  the  bank  were  formed 
nearly  horizontal  cracks  also  appeared  in  the 
cut  near  the  bottom  of  the  bank,  but  which 
of  these  were  the  earlier  it  seemed  impossible 
to  decide. 

After  a  break  has  made  a  fair  start  the  cracks 
more  remote  from  the  cut  gape  and  show  under- 
lying curved  surfaces  which  reach  the  general 
level  of  the  top  of  the  bank  nearly  at  right 
angles  or  crop  out  almost  vertically,  and  at  the 
outcrop  the  vertical  cross  section  of  such  a 
surface  shows  a  very  moderate  radius  of  curva- 
ture. The  surfaces  of  rupture  are  fairly 
smooth,  many  of  them  slickenslided  a  little 

2  Mr.  MacDonald  records  that  "some  of  the  blocks  sank  a  little  in  front 
and  tilted  up  in  the  rear,  so  that  they  were  a  yard  above  the  front  part 
of  the  block  behind."  This  behavior  was  unusual,  and  I  saw  no  in- 
stances of  it.  Local  inhomogeneities  in  the  bank  might  perhaps  bring 
about  irregularities  in  surfaces  of  rupture  which  would  account  for  excep- 
tional throws  of  2  or  3  feet.  No  other  suggestion  on  this  subject  occurs 
to  me. 

253 


254 


SHORTER  CONTRIBUTIONS   TO   GENERAL   GEOLOGY,  1916. 


below  the  outcrop,  but  not  smooth  enough  to 
make  accurate  measurements  of  their  radii  of 
curvature  practicable.  As  nearly  as  I  could 
discover  these  radii  measured  between  100  and 
200  feet.  Where  these  underlying  ^urf aces  are 
exposed  to  a  considerable  extent  it  is  apparent 
that  the  radii  of  curvature  increase  rapidly 
with  increasing  depth,  and  some  exposures 
from  which  disintegrated  material  had  been 
removed  appeared  to  prove  that  as  the  cut  is 
approached  the  radius  of  curvature  becomes 
very  large  indeed. 

Movement  of  the  slides  perhaps  never  en- 
tirely ceases,  but  it  varies  greatly  in  velocity, 
from  a  fraction  of  an  inch  a  day  to  many  yards. 
After  considerable  motion  has  taken  place  the 
sheets  of  rock  are  broken  up  and  the  external 
surface  of  the  slide  becomes  as  rough  as  a 
choppy  sea. 

A  certain  amount  of  consolidation  and  of 
what  might  be  called  secondary  cohesion  some- 
times occurs  in  a  slowly  moving  slide  of  large 
dimensions  after  the  material  has  been  reduced 
to  a  chaotic  condition.  In  such  cases  well- 
developed  curved  surfaces  of  rupture  and  step 
faults  form,  indistinguishable  in  general  char- 
acter from  the  initial  disturbances  in  the  solid 
bank.  This  surprising  fact  indicates  that 
definite  mechanical  laws  of  wide  applicability 
underlie  the  formation  of  slides.  I  was  witness 
to  these  phenomena  in  the  Cucaracha  slide, 
and  they  have  made  their  appearance  in  other 
and  more  recent  breaks. 

During  the  progress  of  a  large  slide  upheaval 
of  the  bottom  of  the  canal  may  take  place  from 
time  to  time,  showing  that  deformation  of  the 
rocks  extends  to  a  certain  depth  below  the 
deepest  excavation ;  but  this  upheaval  does  not 
attend  every  spasm  of  activity  in  the  slide, 
nor  does  the  amount  of  material  thrust  up  in- 
dicate that  deformation  extends  more  than  a 
few  yards  beneath  the  bottom  of  the  canal. 
A  layer  of  rock  say  a  hundred  feet  in  width, 
buckled  by  nearly  horizontal  pressure,  would 
show,  even  if  it  were  only  a  couple  of  yards  in 
thickness,  mounds  of  rubble  as  much  as  20  or 
30  feet  in  height,  or  of  the  order  of  magnitude 
of  the  observed  upthrusts. 

LIMITING  DEPTH  OF  DISTURBANCE. 

To  simplify  the  mechanical  problem  as  much 
as  possible,  suppose  the  case  of  a  level  plain 
underlain  to  a  great  depth  by  an  ideally  ho- 


mogeneous rock.  At  any  depth  in  this  rock 
the  pressure  will  be  hydrostatic  and  equal  to 
the  depth  multiplied  by  the  density.  Suppose 
a  narrow  trench  to  be  sunk  vertically  in  this 
rock,  the  width  being  so  small  that  caving  of 
the  sides  can  be  prevented  by  mine  timbering. 
Then,  because  of  the  one-sided  relief  of  pressure 
there  will  be  at  the  bottom  of  the  cut  a  hori- 
zontal stress,  directed  from  the  wall  into  the 
cut,  which  is  equal  to  the  product  of  the  depth 
and  the  density.  This  stress  will  tend  to  pro- 
duce a  horizontal  shear  and  to  drive  the  bot- 
tom of  the  wall  into  the  cut.  If  the  cut  is 
sunk  deep  enough,  so  deep  that  the  stress  is 
equal  to  the  resistance  of  the  rock  to  simple 
shearing  stress  at  the  elastic  limit,  this  defor- 
mation will  occur  and  the  wall  will  bulge. 

This  seems  a  rather  hasty  statement,  but  in 
the  last  section  of  this  paper  the  strains  are 
considered  in  detail;  it  is  there  shown  that  the 
elastic  limit  for  simple  shear  would  be  reached 
long  before  the  limit  for  mere  linear  compres- 
sion, and  that  of  all  elementary  resistances  that 
resistance  which  opposes  stress  such  as  is  ex- 
erted by  a  pair  of  scissors  is  the  weakest. 

Let  the  limiting  depth  at  which  this  one 
species  of  flow  makes  its  appearance  be  denoted 
by  VD  so  that  if  >  'is  the  density  the  hydro- 
static pressure  is  pyl}  which  is  also  the  value 
of  the  shearing  stress. 

CONDITIONS  IN  A  WIDE  CUT. 

The  hypothesis  of  a  narrow  timbered  cut 
was  employed  in  finding  the  limiting  depth,  yl} 
in  order  to  avoid  the  complication  of  a  caving 
bank.  Let  a  wide  cut  be  substituted,  one  a 
mile  wide  if  the  reader  chooses,  but  let  the  bank 
be  vertical.  Then  even  before  the  depth  yt  is 
attained  any  real  rock  wall  would  break  down 
or  cave.  But  imagine  for  a  moment  the  rock 
replaced  by  a  substance  so  tough  that,  though 
it  would  undergo  permanent  deformation  at  the 
same  limit  as  the  rock,  it  would  hang  on  long 
enough  to  be  studied.  A  ductile  substance, 
such  as  wrought  iron,  would  act  in  this  way. 

Consider  a  surface  of  uniform  deformation 
nearly  as  deep  as  y^  and  extending  into  the 
wall.  This  surface  will  surely  not  be  horizon- 
tal, for  such  a  strain  would  imply  the  expendi- 
ture of  an  infinite  amount  of  energy. 

Before  caving  can  take  place  in  a  homogene- 
ous bank  the  material  of  the  bank  must  be 
strained  to  its  elastic  limit.  The  vertical  cross 


MECHANICS   OF    THE   PANAMA   CANAL  SLIDES. 


255 


section  of  the  bank  must  therefore  include  a 
line  along  which  the  strain  is  uniform.  This 
line  must  reach  the  top  of  the  bank  somewhere, 
and  it  may  be  assumed  that  the  line  is  curved, 
because  that  is  a  far  more  general  hypothesis 
than  that  it  is  straight,  besides  being  in  har- 
mony with  observation. 

In  fig.  22  OBC  represents  the  bank  and 
ABOD  a  part  of  the  cut.  The  x  axis,  or  OX,  is 
taken  at  a  depth  yt  from  the  original  surface, 
and  EC  is  a  curved  line  along  which  the  shearing 
stress  is  uniform.  The  problem  is  to  find  its 
equation. 

At  any  point  the  original  hydrostatic  pressure 
was  (]/i  —  y)p,  but  excavation  of  the  cut,  having 
disturbed  the  original  equilibrium  and  brought 
about  strain  reaching  the  elastic  limit,  has 
developed  a  shearing  stress  which  is  equal  to 


FIGURE  22.  —  Curve  of  uniform  tangential  strain. 

(y\~y}p  Per  unit  length  and  which  is  of  itself 
inadequate  to  cause  flow.  But  there  is  another 
manifestation  of  stress  to  be  considered.  The 
shearing  stress  is  equivalent  to  a  tension  in  the 
direction  of  the  curve,  say  T  per  unit  length. 
Let  ^  be  the  angle  which  the  tangent  to  the 
curve  makes  at  xy;  let  8$  be  an  elementary 
angle  and  8s  a  corresponding  arc.  Then  ele- 
mentary mechanics  shows  that  the  tension,  T, 
acting  along  the  arc  5s  is  equivalent  to  a 
normal  pressure  1  T8\f/. 

It  has  already  been  explained  that  pi/t  per 
unit  length  is  the  shearing  stress  needful  to 
strain  the  mass  to  its  elastic  limit  for  simple 
shear.  Hence  if  stress  of  this  intensity  is  to  be 
set  up  along  the  curve  EC  the  following  equa- 
tion must  hold  good: 


or,  more  briefly, 

T      8s 


i  See  Tait,  P.  Q.,  Properties  of  matter,  p.  253,  1894;  or  Lamb,  H., 
Statics,  p.  276,  1912. 


Here  8s/8\f/  is  the  radius  of  curvature,  say  R, 
while  Tfp  is  a  constant  characteristic  of  the 
material  and  essentially  positive.  It  may 
therefore  be  replaced  by  62,  and  then 


which  is   the  most  general   equation   of   the 
elastic  curve. 

Replacing  R  by  its  value  in  terms  of  dy/dx 
and  d2y/dx2  and  integrating  once  gives 


=  C—2l2  cos 


(1) 


where  C  is  a  constant  of  integration.  The 
form  of  the  curve  depends  on  the  value  of  C. 
For  the  present  problem  it  is  evident  that  the 
curve  can  not  cross  the  x  axis  and  that  y  can  not 
become  negative,  so  that  C  must  equal  or  ex- 
ceed 2&2.  It  is  easily  proved  that  if  C=  2b2  the 
equation  represents  a  curve  coinciding  with  the 
x  axis  for  an  infinite  distance.  This  is  not  a 
case  to  be  considered,  and  therefore  C>2b~. 
The  equation  then  represents  the  elastic  curve 
of  Euler's  eighth  class,  a  diagram  of  which  is 
given  in  Thomson  and  Tait's  "Natural  philos- 
ophy," §  611,  figure  7. 

For  some  purposes  equation  (1)  is  conven- 
ient enough.  Thus  if  the  ordinate  of  the  point 
at  which  the  tangent  of  the  curve  is  vertical  is 
called  T/V,  then  O=  yv2 ;  while  if  the  ordinate  at 
the  point  where  the  tangent  is  horizontal  is  y0, 
then  i/02  =  yv2  —  2i2.  But  values  of  the  abscissae 
are  not  so  simple. 

It  is  needless  to  say  that  the  geometry  of  the 
elastic  curve  has  been  thoroughly  known  for  a 
century  and  that  this  is  no  place  to  expound 
the  subject.  A  few  results,  however,  must  be 
set  down.  By  substituting 


where  <p  is  a  variable  angle  and  Tc  is  the  sine  of 
a  constant  angle,  it  will  be  found  that 


Then  also 


dx  =  cot  ifsdy  =  cot  2<pdy 


(2a) 


and  x  takes  the  form  of  an  elliptic  integral. 

For  purely  practical  reasons  (the  scope  of 
tables  of  elliptic  integrals)  it  is  convenient  to 
reckon  x  negatively,  or  to  the  left  of  the  origin 


256 


SHOETEE   CONTBIBUTIONS  TO   GENEBAL  GEOLOGY,  1916. 


in  figure  23.    Then,  in  the  conventional  termi- 
nology of  these  integrals,1 


If,  for  example,  the  horizontal  distance  from 
the  origin  to  the  vertical  tangent  of  the  curve 
is  required,  \f/=  —  ir/2  and  <p=  x/4,  and  the  value 
of  x  can  be  computed  from  tables  of  elliptic  in- 
tegrals. Because  of  symmetry,  positive  values 
of  x  have  the  same  absolute  value  as  negative 
values.2 

The  element  of  area  of  the  curve,  ydx,  is 
independent  of  Tc: 


ydx  =  4b2  (sinV  —  — )  d<p  =  b2  cos  \{sd\l/ 


so  that 


ydx  =  b2 


(4) 


a  result  which  can  be  found  directly  from  (1) 
and  (2a). 

The  length  of  the  curve  counted  from  the 
horizontal  point  is  given  by 


.(5) 


and  s/b  is  thus  simply  proportional  to  the  first 
part  of  the  value  for  —  x/b. 

It  should  be  remarked  that  b2  is  an  absolute 
constant  dependent  only  on  the  density  and 
tenacity  of  the  rock,  so  that  geometrically  b  is 
the  unit  in  which  lengths  are  computed  and  b2 
the  unit  area.  On  the  other  hand,  fc  varies 
from  curve  to  curve  of  a  family  of  curves,  all 
of  which  share  a  common  value  of  b,  but  as  fc 
is  the  sine  of  an  angle  it  can  not  exceed  unity. 

LIMITING  VALUES  OF  k. 

It  has  already  been  pointed  out  that  if 
C=  2b*  the  elastic  curve  is  a  horizontal  straight 
line  coinciding  with  the  x  axis.  The  same 
equality  implies  that  Tc  is  unity,  and  therefore, 
for  the  problem  under  discussion,  Ic  must  al- 
ways be  the  sine  of  an  angle  less  than  v/2.  It 

1  For  the  meaning  of  the  symbols  in  equation  (3),  see  for  example 
I'eirce's  "Short  table  of  integrals." 

*  Equation  (3)  is  substantially  identical  with  that  given  by  Lamb 
(Statics,  p.  279),  who,  however,  takes  the  origin  at  a  different  point, 
making  z  and  <p  disappear  together,  so  that  the  y  axis  includes  the  maxi- 
mum value  of  y.  In  (3)  the  origin  is  so  transposed  that  x  and  ^  disap- 
pear together,  so  that,  as  required  for  the  problem  in  hand,  the  y  axis 
passes  through  the  minimum  value  of  y,  or  the  point  for  which  <p=-rft. 


is  equally  evident  that  Jc  can  not  vanish,  for 
were  it  to  do  so  the  curve  would  intercept  the 
vertical  axis  at  an  infinite  distance.  There  are 
other  reasons  for  supposing  that  &  can  not  be 
very  small,  and  these  can  be  very  briefly  stated. 
In  this  discussion  it  has  not  been  needful  to 
consider  any  strains  except  those  at  the  elastic 
limit,  but  the  general  theory  of  elastic  strains 
shows  that  at  the  edge  of  a  vertical  cliff  or 
bank  there  will  be  no  strain  at  all,  and  for 
'some  distance  from  such  an  edge  the  strains 
will  be  exceedingly  small.  Hence  strains  reach- 
ing the  elastic  limit  are  not  to  be  considered 
near  this  edge.  It  might  be  possible,  but  it 
would  not  be  worth  while,  to  determine  how 
near  to  this  edge  the  elastic  limit  could  be 
reached. 

On  the  other  hand,  it  is  very  important  to 
consider  how  far  back  a  curve  of  critical  shear 
can  reach,  and  this  I  believe  to  be  a  simple 
problem.  From  the  manner  in  which  the  equa- 
tion of  the  elastic  curve  was  derived  it  is  appar- 
ent that  the  pressure  due  to  tension  is  a  second- 
ary phenomenon  due  to  elastic  strain.  It  is 
unthinkable  that  this  part  of  the  pressure 
should  exceed  the  whole  pressure  requisite  to 
produce  flow.  But  when  the  curve  crops  out 
on  the  bank  at  90°  to  the  horizontal,  the  pres- 
sure due  to  tension  at  the  outcrop  exactly 
equals  the  critical  tension,  y^p.  Hence  for  a 
given  value  of  yl  the  lowest  possible  curve  is 
that  which  intersects  the  level  bank  at  right 
angles.  From  this  condition  the  maximum 
value  of  h  cap  be  determined. 

•EXAMPLES  OF  SLIDE  CURVES. 

In  order  to  illustrate  conditions  resembling, 
to  a  first  approximation,  those  met  with  in  the 
Culebra  Cut,  I  have  computed  a  few  values  of 
the  more  important  elements  of  the  curves,  and 
these  are  tabulated  below.  It  is  easy  to  see 
that  only  relatively  large  values  of  Ic  =  sin  a  are 
of  interest  q,nd  I  have  begun  with  «=  75°. 
Taking  xl  as  the  abscissa  of  the  vertical  tan- 
gent, it  is  found  from  equation  (3),  while  if  y0  is 
the  value  of  |he  ordinate  for  x  =  o,  yjb  =  2  cot  a. 
Then  y^/b2  =^y02/b2  +  2.  The  fundamental  rela- 
tion y/b  =  &/$j  makes  it  easy  to  find  the  radii  of 
curvature  answering  to  xlyl  and  x0y0.  For  the 
purpose  of  tlie  diagram  it  is  not  requisite  to 
compute  othler  points;  after  describing  an  arc 
at  the  axis  of  symmetry  with  RJb  and  a  second 


MECHANICS  OF   THE   PANAMA   CANAL   SLIDES 


257 


arc  at  x^y^  with  RJb,  the  two  can  be  connected 
without  serious  error  by  the  help  of  a  curved 
ruler. 

Points  on  the  elastic  curve. 


k 

*/| 

Volb 

yjb 

Ro/b 

Rilb 

sin  75°  

1.  3411 

0.  5358 

1.512 

1.  866 

0.  661 

sin  80°  

1.  7094 

.  3526 

1.458 

2.836 

.686 

sin  85°.. 

2.  3728 

.1750 

1.425 

5.714 

.702 

sin  89°  

3.  9690 

.0350 

1.415 

28.  570 

.707 

sin  90°  .  . 

0 

1.414 

.707 

To  estimate  an  appropriate  value  for  b  it  is 
requisite  to  adopt  some  value  for  the  resistance 
of  the  rock  either  to  shearing  stress  or  to  crush- 
ing. The  ultimate  resistance  to  crushing  of 
such  materials  as  soft-burned  brick,  inferior 
concrete,  and  the  poorest  sandstones  is  some- 
what less  than  3,000  pounds  per  square  inch. 


FIGURE  23.— Elastic  curve  for  a=75°,  80°,  85 


The  Cucaracha  formation  is  probably  of  similar 
strength,  and  I  will  assume  its  resistance  to  be 
2,760  pounds.  In  the  concluding  section  of 
thisjpaper  reasons  are  given  for  supposing  that 
6-y/2  times  the  resistance  to  shear  is  about 
equal  to  the  resistance  to  crushing,  and  this 
implies  that  for  the  Cucaracha  the  resistance  to 
shear  is  325  pounds  per  square  inch.  This  is 
the  weight  of  a  column  of  rock  of  a  density  2.5 
times  that  of  water  and  300  feet  high. 

If  the  curve  for  which  a  =  89°  is  selected  and 
7/1  is  taken  as  300  feet, 


By  multiplying  all  the  lengths  given  in  the  table 
by  212  a  consistent  set  of  values  is  obtained. 

In  the  diagram  (fig.  23)  the  height  of  the 
bank  above  the  x  axis  is  taken  as  300  feet  and 
the  curve  for  a  =  89°  cuts  it  perpendicularly 
at  a  distance  of  841  feet  from  the  y  axis.  The 


curves  for  smaller  values  of  a  give  larger  values 
for  yl  and  therefore  cut  the  300-foot  level  at 
acute  angles. 

According  to  the  theory  here  set  forth,  a  limit 
is  set  to  the  vertical  height  of  a  cliff  or  any  rock. 
From  results  obtained  by  the  United  States 
Geological  Survey  it  appears  that  granites 
show  resistances  up  to  34,000  pounds  per  square 
inch,  which  would  correspond  to  a  cliff  3,700 
feet  high.  The  brow  of  El  Capitan,  in  the 
Yosemite  Valley,  stands  3,100  feet  above  the 
valley,  but  the  top  of  the  dome,  some  2,000 
feet  back  from  the  brow,  is  about  500  feet 
higher. 

HYDROSTATIC  ANALOGY. 

If  two  rectangular  blocks  of  very  clean  glass 
are  placed  in  a  dish,  parallel  to  one  another, 
and  if  water  is  added  until  the  faces  of  the 
blocks  nearest  together  are  wet  to  the  top  in 
consequence  of  capillarity,  then  the  vertical 
cross  section  of  the  water  surface  between  the 
blocks  is  the  elastic  curve  represented  by 
equations  (2)  and  (3) ;  the  height  of  the  blocks 
above  the  general  water  level  is  given  by  yj), 
and  the  amount  of  water  raised  above  this  level 
by  capillarity  or  surface  tension  is  62  per  unit 
length  for  each  wall  of  the  channel  between 
the  blocks.  If  the  surface  tension  is  T  and 
the  density  is  p  then  T/p  =  62. 

This  system  is  in  stable  equilibrium,  the 
surface  of  the  water  is  minimal  for  the  bound- 
ary conditions,  and,  as  the  equilibrium  is  stable, 
the  gravitational  potential  is  a  minimum. 
The  whole  system  may  be  supposed  solidified 
without  disturbance  of  equilibrium.  One-half 
of  this  model,  to  the  right  or  the  left  of  the 
point  at  which  the  capillary  curve  is  lowest, 
represents  the  mass  beneath  a  slide  on  the 
Culebra  Cut.  The  whole  model  represents  the 
slide  surfaces  as  they  would  be  were  the  cut 
extremely  narrow,  provided  that  the  material 
sliding  in  were  removed  as  fast  as  it  came  until 
the  slides  "died." 

This  very  perfect  analogy  and  the  theory  of 
this  paper  seem  to  me  to  show  that  the  profile 
of  the  bed  or  bottom  of  a  straight  watercourse 
or  river,  flowing  through  a  homogeneous 
stretch  of  country,  must  tend  to  approach  the 
elastic  curve,  and  that  this  profile  is  also  most 
suitable  for  a  canal. 


258 


SHORTER  CONTRIBUTIONS   TO   GENERAL   GEOLOGY,  1916. 


FORMATION  OF  RUPTURES. 

Thus  far  the  discussion  has  been  limited  to 
conditions  appropriate  to  incipient  flow;  the 
rock  has  been  supposed  strained  to  its  elastic 
limit,  but  short  of  the  point  of  rupture.  In 
such  materials  as  rocks,  which  are  to  be  classi- 
fied as  brittle  substances,  the  difference  of 
stress  between  the  so-called  limit  of  solidity 
and  the  breaking  point  is  extremely  small. 
Moreover,  as  a  matter  of  course  real  rocks  are 
not  homogeneous. 

Suppose  that  the  limit  of  solidity  has  been 
exceeded  by  a  minute  stress  increment,  but 
that  along  some  small  arc  of  the  elastic  curve 
the  rock  were  more  brittle  than  elsewhere: 
then  evidently  a  local  crack  would  develop; 
the  resistance  along  the  entire  curve  would  be 
diminished  pro  tanto;  the  stress  on  the  re- 
maining larger  portion  of  the  curve  would  be 
correspondingly  increased;  further  rupture 
would  follow;  and,  as  it  appears  to  me,  the 
crack  would  extend  from  one  end  of  the  curve 
to  the  other  hi  much  less  time  than  is  required 
to  state  this  conclusion.  So,  on  a  frozen  lake, 
when  a  sudden  fall  of  temperature  occurs,  a 
crack  starts  with  a  report  at  some  point  along 
the  shore  and  tears,  booming,  across  the  ice 
sheet  at  a  velocity  approaching  that  of  sound. 

If  before  rupture  there  is  plastic  flow  along 
a  given  curve,  then  after  rupture  the  overlying 
mass  can  move  by  gravity;  for  till  rupture 
occurred  motion  was  opposed  by  cohesion,  and 
when  this  is  overcome  resistance  is  diminished. 
Thus  there  is  a  surplus  of  energy  available  to 
accomplish  work. 

BULGING  OF  CANAL  BOTTOM. 

The  necessary  and  sufficient  condition  for 
flow  is  that  Ry  =  ~bz,  and  the  smallest  value 
which  R  can  reach  is  R1  =  b2/yl.  The  stresses 
which  bring  about  this  condition  are  due  to 
the  tendency  of  the  cliff  to  settle  down  into 
the  cut,  and  this  tendency  will  persist  until 
flow  takes  place  along  the  basal  curve  for  which 
^  =  7r/2  at  the  outcrop. 

Strain  can  not  be  confined  to  levels  above 
the  bottom  of  the  cut,  for  the  moment  the  bank 
begins  to  sag,  even  within  the  elastic  limit,  ad- 
joining masses  are  stressed  to  some  extent,  and 
these  stresses  must  extend,  with  diminished 
intensity,  to  great  distances.  Thus  even  while 
the  cut  is  shallow  there  must  be  elastic  strains 


along  the  basal  curve.  As  th:  depth  of  the 
cut  increases  the  strain  along  this  curve  must 
increase  until  it  approaches  the  elastic  limit, 
both  in  the  wall  and  below  the  cut  in  the  plane 
of  the  wall. 

Now  suppose  that  the  cut  is  nearly  but  not 
quite  down  to  the  basal  curve  and  that,  by 
some  local  inequality  in  the  resistance  of  the 
material  on  some  part  of  the  basal  curve,  or  in 
consequence  of  some  jar,  due  perhaps  to  move- 
ments in  the  bank,  a  short  local  crack  forms  on 
some  part  of  the  basal  curve :  then  the  question 
arises  whether  or  not  this  crack  will  spread. 
Movement  of  the  mass  overlying  the  curve  will 
be  opposed  by  the  horizontal  resistance  to 
crushing  or  buckling  of  the  mass  underlying 
the  floor  of  the  cut  and  extending  down  to  the 
curve;  but  when  this  stratum  has  been  reduced 
to  a  very  small  thickness  the  crack  may  extend 
and  cross  the  vertical,  thus  splitting  off  a  layer 
of  rock  immediately  beneath  the  cut.  As  has 
been  pointed  out  above,  the  formation  of  a 
crack  along  the  curve  suddenly  releases  an 
amount  of  the  energy  of  position  of  the  bank 
corresponding  to  the  cohesion  which  existed 
until  the  crack  formed  and  spread.  At  the  ex- 
pense of  this  energy  buckling  or  bulging  of  a 
thin  bottom  layer  may  take  place. 

This  seems  to  me  an  adequate  qualitative 
explanation  of  the  upheavals  of  the  floor  of 
the  cut  observed  during  the  later  part  of  the 
excavation.  That  shock  had  something  to  do 
with  these  upheavals  is  suggested  by  the  fact 
that  continuous  slow  upheavals  corresponding 
to  the  slower  movements  of  the  slides  were  not 
observed.  Upheavals  accompanied  only  the 
spasmodic  accelerations  of  slide  movement. 
This  phenomenon  is  a  harbinger  of  what  would 
occur  if  the  cut  were  extended  down  to  the  full 
depth  ylf  for  then  the  bottom  and  sides  of  the 
cut  would  ooze  in  continuously  by  plastic  flow. 

EFFECT  OF  THE  FORM  OF  THE  BANKS. 

To  simplify  discussion  it  has  been  assumed 
that  the  canal  was  a  vertical  cut  through  a  flat 
country  underlain  by  homogeneous  rock,  and 
of  course  these  assumptions  are  not  in  accord 
with  the  facts.  But  the  country  is  rather  flat; 
and  as  the  underlying  rock  is  a  solid  mass, 
though  not  a  strong  one,  the  variability  of 
load  near  the  surface  must  be  fairly  well  dis- 
tributed at  depths  of  more  than  100  feet. 


MECHANICS   OF    THE   PANAMA   CANAL   SLIDES. 


259 


Until  slides  began  to  give  trouble  the  banks 
of  the  cut  were  very  steep — quite  too  steep,  in 
fact,  as  everyone  would  now  concede.  It  is 
well  to  consider  what  would  have  been  the  ef- 
fect of  giving  the  excavation  lower  slopes. 

The-ordinary  theory  of  earth  pressures  on 
retaining  walls  is  based  on  the  existence  of  an 
angle  of  rest  in  a  pile  of  discrete  particles.  It 
appears  to  me  to  be  totally  inapplicable  to  con- 
ditions in  the  Cucaracha  formation,  for  the 
mere  existence  of  breaks  demonstrates  that  the 
mass  possesses  continuity.  The  rocks  of  the 
Culebra  Cut  behave  very  much  as  a  mass  of 
agar-agar  jelly  might  do  if  a  rectangular  mold 
of  this  substance,  a  foot  or  so  in  depth,  were 
turned  out  on  a  horizontal  table.  If  the  jelly 
were  of  the  right  degree  of  stiffness,  the  edges 
of  the  mass  would  first  sag,  and  then  breaks 
would  make  their  appearance;  but  nothing  re- 
sembling a  constant  angle  of  rest  would  be 
developed.  To  work  out  a  complete  theory  of 
the  relief  of  pressure  in  such  a  jelly,  or  in  the 
Cucaracha  formation,  due  to  an  inclination  of 
the  walls,  would  probably  be  very  difficult. 
Nevertheless,  very  simple  considerations  show 
that  sloping  the  walls  is  an  effectual  method 
of  reducing  the  pressure. 

If  the  Culebra  Cut  were  replaced  by  an  ex- 
ceedingly strong  wall,  the  pressure  against  the 
wall  would  be  hydrostatic.  For  a  small  change 
of  depth  the  increment  of  pressure  would  be 
p(yl  —  y')d(yl  —  y},  and  the  whole  horizontal 
pressure  from  the  surface  to  depth  yi  —  y0 

would  be  |(?/1-'?/o)2- 

Now,  imagine  a  plane  inclined  to  the  hori- 
zon at  45°  and  passing  through  the  point 
x  =  o,  y  =  yQ.  This  plane  would  cut  off  a 
triangular  slab,  say  of  unit  thickness  and  of 
mass  w  =  \  p(yl  — 1/0)2. 

The  amount  of  frictional  resistance  depends 
primarily  upon  normal  pressure,  so  that  if  F  is 
the  frictional  resistance  and  N  the  normal 
pressure 

F 
_=/i 

where  n  is  the  coefficient  of  sliding  friction  and 
&  the  angle  of  friction.  Now,  F  can  not  exceed 
the  normal  pressure  N,  which  excites  it,  so 
that  ju,  can  not  exceed  unity  and  #  can  not  ex- 
ceed 45°.  Hence  friction  can  not  prevent 


movement  on  a  slope  of  45°.  Thus  if  the  tri- 
angular mass  of  rock  (or  of  jelly)  were  actually 
separated  from  the  remainder  of  the  mass,  fric- 
tion would  not  prevent  it  slipping  down  the 
steep  slope.  The  tangential  pressure  which  the 
mass  w  would  exert  on  the  45°  plane  would  be 
w/V2,  and  this  would  be  resolved  into  a  ver- 
tical pressure  and  a  horizontal  pressure  each 
equal  to  w/2. 

Thus  of  the  whole  hydrostatic  horizontal 
thrust  exerted  against  the  vertical  wall,  just 
one-half  is  exerted  by  the  triangular  slab. 
Hence  also  sloping  the  bank  of  a  cut  at  45° 
would  diminish  the  horizontal  thrust  to  one- 
half  of  its  maximum  value. 

It  is  not  difficult  to  perceive  by  the  further 
application  of  elementary  statics  that  the 
thrust  would  be  still  more  diminished  by  mak- 
ing the  slope  smaller  than  45°. 

The  precaution  of  giving  the  banks  a  low 
slope  might  have  prevented  the  occurrence  of 
slides,  but  as  a  remedial  measure,  after  breaks 
have  developed  to  a  considerable  extent,  it 
seems  to  me  of  little  avail.  After  the  basal 
curve  has  developed  into  a  crack,  the  material 
overlying  it  is  either  in  motion  or  in  unstable 
equilibrium;  and  sooner  or  later  all,  or  nearly 
all  of  it,  will  reach  the  bottom.  Slides  of  origin 
similar  to  those  of  the  Culebra  Cut  are  by  no 
means  confined  to  the  Canal  Zone.  In  my 
opinion  banks  of  cuts  should  be  watched  with 
extreme  care,  and  the  moment  any  cracks  make 
their  appearance  all  other  work  should  be  sus- 
pended until  a  safe  slope  has  been  established. 
Breaks  should  be  prevented,  because  they  can 
not  be  cured. 

NOTE  ON  FINITE  STRAINS. 

Plastic  flow  is  continuous  deformation  with- 
out change  of  density.  It  takes  place  at  the 
so-called  limit  of  solidity.  During  flow,  there- 
fore, a  solid  must  be  treated  as  compressed  to 
a  constant  extent,  and  as  the  elasticity  of 
volume  is  perfect,  when  stress  is  relieved  the 
original  volume  is  restored.  In  nearly  all  cases 
a  solid  mass  undergoing  flow  is  to  be  treated  as 
incompressible . 

This  limit  of  solidity  depends  on  the  type  of 
strain  to  which  the  mass  is  subjected  and  to 
some  extent  on  viscosity.  It  would  also  de- 
pend on  heterotropy,  but  this  paper  deals  only 
with  isotropic  matter. 


260 


SHORTER   CONTRIBUTIONS   TO   GENERAL   GEOLOGY,  1916. 


In  any  strain  ellipsoid  there  are  two  sym- 
metrically oriented  sets  of  planes  of  maximum 
tangential  strain  or  maximum  slide.  If  the 
strain  is  a  rotational  one  (so  that  the  groups  of 
material  particles  through  which  the  ellipsoidal 
axes  pass  vary  with  the  progress  of  the  strain), 
then  there  is  a  difference  hi  behavior  of  the  mass 
on  these  two  sets  of  planes.  Along  that  set  of 
geometrical  planes  which  rotates  more  rapidly 
through  the  mass,  or  on  which  the  material 
particles  change  more  quickly,  greater  resist- 
ance is  offered  to  flow  or  rupture  than  on  the 
other  set,  because  the  resistance  to  be  over- 
come is  rigidity  plus  viscosity  and  because 
viscosity  offers  great  resistance  to  a  sudden 
stress  but  very  small  resistance  to  a  stress 
slowly  applied. 

In  one  strain,  called  simple  shear,  shearing 
motion,  slide,  or  scission  by  various  writers } 
there  is  one  set  of  these  planes  which  is  fixed 


their  product  is  constant,  and  if  all  three  diame- 
ters pass  through  the  same  material  particles  at 
all  stages  of  the  strain,  then  this  strain  is  a 
shear,  though  not  a  simple  shear  but  yet  far 
simpler  than  a  simple  shear.  Both  strains  are 
illustrated  in  figure  24. 

A  pure  shear  may  be  conceived  as  the  result- 
ant of  two  scissions  whose  rotations  are  equal 
and  opposite,  a  fact  of  which  use  may  be  made 
in  the  present  discussion. 

If  a  cube  of  homogeneous  isotropic  matter  is 
subjected  to  uniformly  distributed  pressure  on 
two  opposite  faces,  or  if  the  cube  rests  on  a 
rigid  plane  and  carries  a  normal  load  or  initial 
stress,  P,  then,  no  matter  whether  the  load  and 
the  strain  produced  are  infinitesimal  or  finite, 
just  one-third  of  the  load  is  employed  in  pro- 
ducing cubical  compression,  the  remaining 
two-thirds  being  employed  in  producing  two 
pure  shears  at  right  angles  to  each  other. 


Shear,  ratio  % 


FIGURE  24.— Diagram  illustrating  simple  shear  and  shear,  each  of  ratio  5/4.    The  broken  lines  show 
directions  of  maximum  tangential  strain. 


relatively  to  the  material,  while  the  other  set 
of  planes  of  maximum  tangential  strain  changes 
its  position  relatively  to  the  material  particles 
more  rapidly  than  in  any  other  strain. 

In  scission,  therefore,  flow  will  be  more  easily 
produced  on  the  fixed  set  of  planes  than  in  a 
strain  of  any  other  type;  but  on  the  other  set 
of  planes  flow  will  be  less  easily  produced  in 
scission  than  in  a  strain  of  any  other  typo. 
Scission  is  due  to  a  couple  acting  against  a  re- 
sistance. It  is  the  only  strain  produced  in  a 
rod  of  circular  cross  section  when  the  rod  is 
twisted  about  its  axis. 

Irrotational  or  pure  shear,  usually  denoted 
simply  as  shear,  is  the  simplest  conceivable 
deformation.  If  a  sphere  is  so  distorted  that 
one  diameter  retains  its  length  unaltered  while 
two  other  orthogonal  diameters,  in  a  plane 
perpendicular  to  the  first,  are  so  changed  that 


If  the  strain  at  the  elastic  limit  is  small  and 
if  P  just  exceeds  the  initial  stress  needful  to 
produce  this  strain,  the  conditions  for  flow  are 
fulfilled.  But  to  produce  yielding  relative  mo- 
tion must  take  place  parallel  to  four  planes  all 
of  which  are  at  or  very  close  to  inclinations  of 
45°  to  the  direction  of  the  load.  Suppose  that 
there  were  only  four  planes  of  relative  motion, 
each  passing  through  two  opposite  edges  of  the 
cube;  then  if  a  face  of  the  cube  is  assumed  as 
the  unit  area,  the  area  of  each  of  the  planes  of 
relative  motion  will  be  -^2,  and  to  produce  any 
yielding  by  shear  the  total  area  of  relative  mo- 
tion must  be  at  least  4V2^=5.657. 

Now  on  each  of  these  four  surfaces  the  rela- 
tive motion  may  be  conceived  as  due  to  a  scis- 
sion, the  four  rotations  of  the  scissions  annulling 
one  another  by  pairs.  But  if  the  cube  were  cut 
or  permanently  deformed  by  scission  along  a 


MECHANICS    OF    THE    PANAMA   CANAL   SLIDES. 


261 


plane  parallel  to  any  face  the  area  of  deforma- 
tion would  be  only  unity  instead  of  5.657. 

Call  the  load  or  initial  stress,  P,  when  just 
sufficient  to  induce  flow  by  pure  shears  K,  and 
let  Ka  be  the  tangential  stress  needful  to  induce 
incipient  scission.  Then  it  follows  from  the 
reasoning  stated  above  that 


so  that  if  K  is  known  a  rational  estimate  of  Ka 
can  be  made. 

Experimental  data  as  to  the  relative  values 
of  .fiT  and  KB  for  stone,  concrete,  or  brick  are  not 
to  be  had,  so  far  as  I  know;  but  for  these  sub- 
stances the  limit  of  elastic  strain  and  the  break- 
ing point  lie  very  close  together.  According  to 
Bauschinger  the  ultimate  resistance  to  shearing 
of  stone  is  a  thirteenth  of  the  resistance  to 
crushing,  and  this  substantially  coincides  with 
Von  Bach's  result  for  granite.  Thus  experi- 
ment confirms  the  conclusion  that  relatively 
brittle  substances  will  yield  to  shearing  stresses 
very  much  less  intense  than  would  be  needed 
to  produce  flow  by  irrotational  strains. 

As  flow  is  thus  dependent  on  the  type  of 
strain,  it  follows  that  flow  on  one  set  of  planes 


of  maximum  tangential  strain  may  be  accom- 
panied by  no  sensible  plastic  deformation  on 
the  opposite  set  or  may  there  even  be  attended 
by  rupture. 

SUMMARY. 

After  describing  the  essential  features  of  the  breaks  on 
the  Culebra  Cut  the  author  points  out  that  there  is  a  limit 
to  the  depth  of  a  vertical  cut  in  an  homogeneous  isotropic 
mass,  the  upper  surface  of  which  is  plane.  This  limit  ia 
that  at  which  the  pressure  is  sufficient  to  produce  simple 
shear  in  the  mass,  and  in  a  concluding  note  reasons  are 
given  for  believing  that  6-\/2  multiplied  by  the  resistance 
to  such  shear  is  about  equal  to  the  ultimate  strength  under 
linear  compression.  The  depth  at  which  one-sided  relief 
of  pressure  will  produce  simple  shear  is  called  yl. 

It  is  shown  that  in  such  a  bank  the  profile  of  a  surface 
along  which  the  mass  is  strained  to  the  elastic  limit  must 
be  a  form  of  the  elastic  curve,  the  directrix  of  which  lies 
at  a  depth  yv. 

The  lowest  or  basal  slide  curve  is  one  which  intersects 
the  horizontal  bank  at  right  angles.  Examples  are  worked 
out  for  this  and  other  cases. 

A  complete  analogy  exists  between  the  form  of  these 
curves  and  those  which  the  surface  of  water  assumes  when 
it  rises  by  capillarity  between  vertical,  parallel  glass  plates. 

In  view  of  these  results  the  author  discusses  to  some  ex- 
tent the  formation  of  ruptures,  the  bulging  of  the  canal 
bottom,  and  the  effect  upon  pressure  of  the  form  of  the 
banks.  A  note  on  finite  strains  is  placed  at  the  end  of  the 
paper  in  order  to  facilitate  skipping. 


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